ACOUSTIC WAVE PROPAGATION IN AND
AROUND A FLUID-FILLED BOREHOLE OF
IRREGULAR CROSS-SECTION
by
Chengbin Peng 1 and C.H. Cheng
Earth Resources Laboratory
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
Boreholes with 10% or more ellipticity are not uncommon. In this paper, we consider
the coupling of an incident elastic wave into a borehole of irregular cross-section and
investigate the cross-mode coupling phenomenon in sonic well logging in the presence
of borehole irregularity. The mode-matching method is used. Different from its original
formulation, we employ the Reichel et al. algorithm to obtain the discrete least square
approximation by trigonometric polynomials, a technique closely related to the fast
Fourier transform (FFT). Our method not only yields great accuracy but also gains
computational speed. Our study shows that the pressure in the borehole fluid is sensitive
to the irregularity of the borehole cross-section, it is larger if the incident wave is along
the effective minor axis and smaller if the incident wave is along the effective major axis.
In the frequency range of a typical borehole experiment, the solid displacement in the
formation is much less affected by the borehole irregularity. In an elliptical borehole,
a monopole source excites dipole wave trains that are characteristic of the tube waves,
and a centered dipole source excites monopole wave trains that are characteristic of the
flexural waves.
INTRODUCTION
It has been observed that the boreholes drilled into the earth are noncircular and often
of irregular shape (Hilchie et aI., 1968; Dart et al., 1989; among others). Elastic wave
propagation in a noncircular borehole attracted some attention in recent years. For
the low frequency limit, Zimmerman (1986) and Norris (1990) proposed a method to
calculate the tube wave velocity in a borehole of arbitrary cross-section. They found that
the tube wave speed is reduced compared to the one in a circular borehole. Ellefsen
(1990) applied a perturbation method to study higher order mode dispersion in an
elliptical borehole. He noticed that the percent change in phase velocity at any frequency
was small in a slightly irregular borehole. Randall (1991a) calculated the dispersion
Ipresently at the Geophysics Research Department. Bellaire Research Center, Shell Development
Company, P. O. Box 481. Houston TX 77001-0481.
156
Peng and Cheng
curves for modes in a noncircular fluid-filled borehole in elastic formations by using the
boundary integral formulation. He found that the flexural and screw modes split into
two distinct branches delineated by their orientations. Using the same method, Randall
(1991b) and Liu and Randall (1991) computed the dipole sonic logs in an elliptical
borehole.
The goal of this paper is to study the coupling between an incident elastic wave
and a fluid-filled borehole of irregular cross-section, in order to understand the effects
of the borehole irregularity on downhole seismic measurements. We will apply a modematching method. Deviating from the original formulation (Okuno, 1990), we will
employ an algorithm by Reichel et al. (1991) (hereinafter referred to as the Reichel et
al. algorithm in the text). This algorithm solves a discrete least squares approximation
by trigonometric polynomials, a technique closely related to the fast Fourier transform.
Compared to the boundary integral formulation (Randall, 1991a, 1991b), our method
is faster because the number of unknowns to be determined is decreased by an order
of magnitude. The method proposed in this paper will also be applied to compute
synthetic sonic logs in a borehole of an irregular cross-section. In the latter application,
multiple sources and arrays of multipole receivers will be used. The synthetic logging
seismograms will be computed to study the cross-mode coupling phenomenon in the
presence of irregular cross-section of the borehole, which may be mistakenly attributed
to formation anisotropy (Hatchell and Cowles, 1992).
This paper includes four sections. In the first section, we will give a theoretical
formulation for elastic wave scattering in a fluid-filled borehole, using the mode-matching
method as well as the Reichel et al. algorithm. We will also address issues concerning
its implementation on a computer. In the next two sections, we will present some
numerical examples of elastic wave coupling and synthetic sonic logging seismograms in
a noncircular borehole, respectively. In the last section, we will give a brief discussion
and draw some conclusions.
THEORETICAL FORMULATIONS FOR ELASTIC WAVE
PROPAGATION IN IRREGULAR BOREHOLES
Consider an elastic plane wave incident upon an infinite fluid-filled borehole, as shown in
Figure 1. The cross-section of the borehole is irregular. The formation is homogeneous
and elastic with compressional wave speed a, shear wave speed [3, and density p. The
borehole is filled with a fluid of density Pf and compressional wave speed af. The radius
of the borehole is parameterized by the following trigonometric series:
L
r( I)) = ro + L(r.(/) cos II) + rb(l) sin W).
(1)
1=1
The objective is to compute the pressure in the fluid and the solid displacements on the
borehole wall for a source in the formation. And conversely, as applications to sonic
well logging, we also compute the full acoustic waveforms for a multiple source in the
borehole.
Acoustic wave propagation
157
Modal Solutions
The general solutions to the displacement potentials in both fluid (<p f) and solid (<p,
and 'l/J ) can be expressed in a cylindrical coordinate system as (Schoenberg, 1986) 2
E
00
<Pf =
O<t':;JW) [A o Jo(k f r)+2 2>n(An cosne+A~ sinne) In(kfr)],
(2a)
n=l
00
<p = - O<~~W) [Eo Hal) (kpr) + 2 2>n(En cosne + E~ sinne) HAl)(kpr)],
(2b)
n=l
V (W) [
, . e) Hn(l)(k)]
E= i(32WS
CoHo(l)()
ksr +2 ~
L..J2.n( Cncosne+Cnsmn
sr,
(2c)
n=l
00
'l/J = - (3~~W) [ -D~ Hal) (ksr) + 2 I:>n(D n sin ne - D~ cosne) HAl) (ksr) ],
(2d)
n=l
where z dependence and time dependence ei(kzz-wt) is assumed. In these equations, kf = Jw 2 /O<J - k}, kp = JW 2 /0<2 - kz2 and k s = Jw 2 /(32 - kz2 are the radial
wavenumbers in the fluid and solid, respectively. The signs of kf, kp, kz are chosen such
that the imaginary part Im(kp, k., kf) ;::: O. V(w) (= _w 2 , without loss of generality)
denotes the source function at a given frequency w.
For this problem, the boundary conditions are the continuities of normal displacement un(r±, e) and normal stress <Tnn(r±, e), and vanishing of tangential stresses <Tnb(r±, e)
and <Tnz(r±,e) at the borehole wall r = r±(e), that is,
ui!) (r-, e) = ui:)(r+, e) + u~) (r+, e),
(3a)
-p(f)(r-,e) = <Ti:J(r+,e) +<T~~(r+,e),
(3b)
<T(s)(r+
e)
nb'
+ <T(i)(r+
e)
nb'
= 0
<Ti:l(r+, e) + <T~l(r+, e) =
l
o.
(3c)
(3d)
In the above expressions, the superscript (i) denotes the quantities associated with
the incident wave, (s) denotes those of the scattered wave in the solid and (f) those in
the fluid. The pressure in the fluid is denoted by p. A local coordinate frame is used
where n is normal to the borehole boundary, b is tangential to the boundary, and z is
in the axial direction (see Figure 1).
The transformation from the cylindrical coordinates (r, e, z) to the local orthogonal
coordinates (n, b, z) on the boundary is accomplished through an angular rotation <p =<
n, r >, which is given by
~f-l I ( -ra(l) sin Ie + rb(l) cos Ie )
ro + ~f=l ra(l) cos Ie + rb(l) sin Ie )'
(
(4)
2No energy from 00 is included in the solution. This may not be true if the borehole cross-section is
highly irregular such that multiply scattered energies travel inward.
158
Peng and Cheng
We transform the displacements and stresses from the cylindrical coordinates (r, 0, z)
to the local orthogonal coordinates (n, b, z) using the following relations:
[ un]
~:
=
[cos~
-s~n~
sin~
cos~
0
n[~~],
and
[ ann anb anz ]
abn abb abz =
a zn
azb
[COS
~
- sin ~
sin~
0]
o
cos~
0
0
CT zz
1
TT aTO. aTZ] [cos ~
[a
aOT aOO aO z
sin ~
aZT a zO a zz
-sin~
0
cos~
0
n·
Upon these rotations, expressions for the normal displacement and tractions on the
boundary are obtained. In terms of harmonic expansions, they are given by:
00
u;fl(r, 0) =
L
in en U;(J(r) [An cos nO + A~ sin nO ] cos ¢
n=O
00
+
L
in en V;(J(r) [ -An sin nO + A~ cos nO ] sin¢,
(5a)
n=O
00
u;:)(r,0) =
L
in enU~¢J(r) [B n cos nO + B~ sin nO ] cos¢
n=O
00
+
L
in en V~¢l(r) [ -Bn sin nO + B~ cos nO ] sin¢
n=O
00
+
L
inenU~<)(r)[Cn cosnO+C~ sinnO] cos¢
n=O
00
+
L
in en V~<)(r) [ -Cn sin nO + C;, cos nO] sin¢
n=O
00
+
L
in en U~I/I)(r) [D n cos nO + D~ sin nO] cos ¢
n=O
00
+
L
in en V~?jJ) (r) [ - D n sin nO + D~ cos nO ] sin ¢,
(5b)
n=O
00
- pU)(r, 0)
= a;(J(r, 0) =
L
in en n;(J(r) [An cos nO + A~ sin nO ],
n=O
00
a(S)
nn (r , 0)
=
L
in en n~¢J(r) [B n cos nO + B~ sin nO] cos2 ¢
n=O
00
+
L
in en T~<P)(r) [ -Bn sinnO + B~ cos nO ] sin2¢
n=O
00
+
L
n=O
in en M~¢) (r) [ B n cos nO + B~ sin nO ] sin2 1J
(5c)
159
Acoustic wave propagation
00
:L in Cn n!p(r) [Cn
+
cosnO+C~ sin nO ] cos 2 ¢
n=O
00
:L in Cn T~e)(r) [ -Cn sin nO + C~
+
cos nO ] sin2¢
n=O
00
:L in Cn M~e)(r) [Cn cos nO + C~
+
sin nO] sin 2 ¢
n=O
00
in cnn~'l/J) (r) [D n cos nO + D~ sin nO ] cos2 ¢
L
+
n=O
00
in Cn T~'l/J)(r) [ -Dn sin nO + D~ cos nO ] sin2¢
L
+
n=O
00
in Cn M~'l/J)(r) [D n cos nO + D~ sin nO ] sin2 ¢,
L
+
(5d)
n=O
O"~~ (r, 0) =
00
[
Lincn
M<¢) (r) - ~¢) (r) ]
n
2
[BncosnO+B~sinnO]sin2¢
n=O
00
+
L
incnT~¢)(r)[-Bn sinnO+B~ cosnO] cos2¢
n=O
00
+
[
L
in en
MC<) (r) _ ~e) (r) ]
n
2
[Cn cos nO + C~ sin nO] sin2¢
n=O
00
+
Lin Cn T~e)(r) [-Cn sinnO+C~ cos nO ] cos2¢
n=O
_ ~'l/J)(r) ]
:L in Cn [M('l/J)(r)
n
2
[ Dn cos nO + D~
00
+
sin nO ] sin 2¢
n=O
00
in Cn T~'l/J)(r) [ -Dn sin nO + D~ cos nO ] cos2¢,
+ L
(5e)
n=O
00
O"~sl (r, 0) = L
in Cn z~¢) (r) [Bn cos nO + B~ sin nO ] cos ¢
n=O
00
+ :Lin cnN~¢)(r) [-Bn sinnO+B~
cos nO ] sin¢
n=O
00
+
L
incnZ!P(r)[Cn cosnO+C~ sinnO] cos¢
n=O
00
+
L
in cnN~e)(r) [ -Cn sin nO + C~ cosnO] sin¢
n=O
00
+
L
.
incnZ~'I/J)(r)[Dn cosnO+D~ sinnO] cos¢
n=O
00
+
:L in cnN~'l/J)(r) [-D n sinnO+D~
n=Q
cosnO] sin</>,
(5£)
160
Peng and Cheng
.
{I ifn=O
(j)
(cP)
where en =
2 if n 2: 1 is the Newmann factor. The radial dependences Un (r),Un (r),
... ,Z~,p)(r),N~,p\r) have been given in Appendix A.
Unlike the case of a circular borehole where the boundary conditions are satisfied by
each individual mode in the harmonic expansions (2a)-(2d), in a noncircular borehole
it is the total field that satisfies the boundary conditions. A least square formalism is
used to determine the unknown coefficients {An,A~, B n , B~, Cn,G~, Dn,D~} such that
the errors in satisfying the boundary conditions are as small as possible.
Fourier Series Expansion
We expand the normal displacements and the normal as well as tangential stresses on
the boundary r+ = r- = r(e) in terms of Fourier series. The results for the normal
displacements are:
00
u;(l(r-, e) = ao +
L: (az cosze + a~sinze),
(6a)
Z=l
00
u;.sl(r+,e) =
bo + L: (bzcosze+b~sinze)
1=1
00
+ co + L: ( Cz cos Ie + c~ sin Ie )
1=1
00
+
do
+ L: ( dz cos Ie + d~ sin Ie ),
(6b)
1=1
where
bz =
Cz =
dl
t
(~.~.1Jzn Cn + !£E t
(b n )zn B n + (b 12
= (d n
) Zn D n
B~j
b~ = (b 21 \n B n + (b 22 )In B~j
C~ j
c~ = ~)In
Cn +
~t C~;
+ (d12 ) In D~j
The elements of matrices all ,a12 , ... ,d 22 are the integrals of the corresponding modal
functions for the displacements along the borehole boundary.
Similarly, the pressure on the fluid side of the borehole wall and the normal as well
as tangential stresses on the solid side of the borehole wall can be written as
00
- p(j)(r-, e) = CT;(,l(r-, e) = eo +
L: ( ez cos ze + e~ sin ze),
£=1
(7a)
161
Acoustic wave propagation
00
(7(8)
nn (r+ , e)
fo
+ I: ( fl cos Ie + ff sin l() )
1=1
00
+
go +
+
ho +
I: (gl cos l() + gf sin Ie)
1=1
00
I: (hi cos l() + h~ sin l() ),
(7b)
1=1
00
(7(8) (r+
nb
e) =
'
00
+ I: (01 cos Ie + 0; sin Ie )
1=1
00
+
Po +
+
qo +
I: (PI cos l() + P; sin l() )
1=1
00
I: (ql cos l() + qf sin l()),
(8a)
1~1
and
00
(7;:] (r+, e)
=
ro
+ I: (rl cos Ie + r; sin l() )
1=1
00
+
80 +
I: (81 cos l() + s; sin l() )
1=1
00
+ to + I: (tl cos Ie + t; sin Ie),
(8b)
1=1
where
fl
ikL En+ ~\n E~;
= (m)ln
91
hi
=
= (h n
Cn + (m)ln
) In Dn +
C~;
(h12 ) In D~;
01
= (on) In
En + (012) In
PI
= (pn \n
Cn +
ql
= ~L
D n + (q12
E~;
ff
= ~)In
En +
~)In E~;
gf = (~~!Jln Cn + ~L C~;
h~ = (h21 ) In Dn + (h22 ) In D~;
O~ = (021) In
En + (022) In
W12 )In C~;
P;
= (21) In
Cn +
L D~;
qf
= ~)In
Dn + (q22 \n
E~;
W22 )In C~;
D~;
~L
En +
~)In B~;
r; =
81
= ~)In
Cn +
~)In C~;
s;
tl
= ~)In
Dn+
~)In D~;
t; = ~)In Dn+ ~L D~.
rl =
~)In
En +
~L E~;
= ~~Jln
Cn+
~)In C~;
The elements of the matrices en ,e12, ... ,t22 are the integrals of the corresponding modal
functions for stresses along the borehoi8'boundary.
162
Peng and Cheng
Least Squares Solution by Reichel et al. Algorithm
In the mode-matching method, a least square formalism is used to match the boundary
conditions between solution regimes, whereby the coefficients in the expansion in terms
of modal solutions are uniquely determined. To this end, we define the mean square
errors on the boundary as
£(U n ) = 1 lui!) (r-, B) - u;.s)(r+, B) - u~)(r+, B)W,
(9a)
11- p(f)(r-, B) - CT~~(r+, B) - CT~~(r+, B)11 2 ,
£(CTnb) = 11- CT~~(r+, B) - CT~l(r+, B)11 2 ,
£(CTnz ) = 11- CT;.sl(r+, B) - CT~f(r+, B)11 2 ,
£(CTnn ) =
where
11/11 2 =
fr
I(s) I(s) ds
=
t"
(9b)
(9c)
(9d)
l(r(B)) l(r(B» r(B) dB
defines the norm of a square integrable function 10 on the boundary. Notice that, in
the above definition, the borehole radius is included as a positive weighting function.
£(u n) is the misfit of the normal displacement on the boundary, £(CTnn ) is that of the
normal stress, and £(CTnb) and £(CT nz ) are those of the two tangential stresses. The
objective is to find appropriate coefficients to minimize all these errors simultaneously3.
We substitute the expressions in equations (4.6) through (4.8) into equations (9a)(9d). For example, we calculate the mean square error of the normal displacement in
equation (9a) and compare it with the minimization problem discussed in Appendix
B. We found that the Reichel et al. algorithm (for least square approximations by
trigonometric polynomials) can be employed to compute rapidly and accurately the
coefficients {aI, a~, bl, bz, ...tl, t/}f=o, where L is the order of truncation. We have assumed
that the displacement and stresses of the incident wave fields are known on the boundary
at a set of distinct sample points. The weights in the Reichel et al. algorithm are chosen
as Wk = rk, corresponding to the radius of the irregular borehole at B = Bk .
Given that {aI, a~, bl, b~, ...tt, taf=o are known, the coefficients {An, A~, B n , B~, Cn, C~,
Dn , D~}:;'=o can then be determined uniquely by solving a (8N + 4)x(8N + 4) matrix
equation4 derived from the relationships given in section 2.2, where N
orders of truncation.
=
L are the
The truncation order is chosen such that the errors in equations (9a)-(9d) are much
smaller than the norms of the incident wave field (relative error < 10-4 ). This guarantees at least four digits of accuracy in the solution. The relative error is dependent
on the shape of the borehole, Le., the spectrum of the borehole cross-section in equation (1). Using double precision computations, we found that the relative error was less
3In the case of a circular borehole, Le., reO) == TO, these errors are minimal if and only if all the
individual modes satisfy the boundary conditions themselves. This is due to the orthogonality of the
modal functions. In this case, the mode-matching method reduces to the method we have developed in
Schoenberg (1986) and Peng et al. (1993) .
. 'lThe total number of unknown coefficients is 8 xN +4, where the 4 is due to the property that
A~=B~=C~=Do=O
Acoustic wave propagation
163
than 10- 14 for a circular borehole and less than 10-9 for an elliptical borehole, where
L = N = 10 was used.
Given the error bounds on the boundary, we can show that the error at any point
away from the boundary is also bounded with
(10)
where M(O) is a positive constant independent of both N (the order of truncation) and
position rEO. Here UN is the calculated displacement by the mode-matching method,
and u is the exact solution. The proof is straight forward by taking account of the
decaying nature of the Hankel functions at large distances.
Implementation
Several issues are of concern in the implementation of this technique. The most important one is the control of accuracy. The sources of inaccuracy may come from the
singularity of the coefficient matrix, causing energy to leak between modes, or from
the loss of significant digits in the evaluation of the Bessel functions In(z) - t 0 and
the Hankel functions H$.I\z) - t 00 as Izi - t 0 and n large, or from the fact that the
borehole cross-section may be so rough that more terms in the summations would be
needed. In Our experience, proper scaling and preconditioning can greatly improve the
condition of the matrix equation. The method we have used to compute the Bessel and
Hankel functions (adapted from du Toit, 1990) is accurate enough for any order and
any complex argument.
Discretization along the borehole boundary also has some effects on accuracy. A
denser discretization is needed at sharp corners, while in flat regions only sparse sampling is necessary. The automatic discretization scheme we use is based on the local
curvature along the boundary. It is combined with the requirement that the size of the
largest element be less than 1/25 of the shortest wavelength. The local curvature in
polar coordinates is given by
r 2 + 2 (dr/d())2 - r d2r/d()2
(11)
K=
(r 2 + (dr/d())2)3/2
In our calculations, the number of samples along the borehole boundary is 180, roughly
in two degree intervals between two successive sampling points.
For a fixed discretization of the boundary, say, 180 elements or grid points, we compare the boundary integral method against the mode-matching method. The boundary
integral method used in Randall (1991a, 1991b) has J = 4x 180 = 720 unknowns to
be determined. This requires arithmetic operations on the order of O(J2) to solve the
matrix equation alone, not including calculations for forming the coefficient matrices.
The mode matching method developed here only needs arithmetic operations on the
order of O(J/4x(8N + 4)) = O(J), where N = 10 is the order of truncation. This
makes the mode-matching method a practical tool of computing synthetic seismograms
in an irregular borehole, where the calculation is repeated for many samples in both
frequency and vertical wavenumber.
Peng and Cheng
164
APPLICATIONS TO BOREHOLE COUPLING
Case of an Elliptical Borehole
Boreholes with 10% ellipticity are not uncommon. Figure 2 shows a borehole cross
section whose shape was determined with a borehole televiewer (Ellefsen, 1991). The
major axis of the borehole is 19.1 cm in diameter; the minor one is 16.9 cm in diameter.
The borehole is discretized with 180 elements where the boundary conditions are satisfied in the least square sense. In the following examples, the frequency of the incident
wave is 1000 Hz. The formation is Berea sandstone. The fluid is ideal water. The order
of truncation is N = L = 10.
Figure 3 shows the relative error in satisfying the boundary condition as a function
of angle of incidence. The error is computed according to
E =
Ilcri(,{ (r-, e) - cr~~(r+, e) - cr~~(r+, e)11
Ilcr~~(r+,e)11
'
(12)
where the norm was defined in section 2.3. For a computation with double precision,
this error is on the order of 10- 9 for this particular example. This result implies that
at least 8 significant digits are obtained in the solution. Surprisingly, the relative error
increases with the increase of the angle of incidence (from 10- 12 at grazing incidence
to 6 X 10-9 at normal incidence). This behavior originates from approximating plane
waves by superposition of finite sets of cylindrical waves.
Another way to demonstrate the accuracy of the mode-matching method is to examine the decay of the coefficients in the expansions (2a)-(2d), or in other words, to
evaluate the contributions of higher order modes to the summations. Figure 4 is a barplot of magnitudes of coefficients {An, En, Cn, Dn}~~b" as a function of harmonic order
n. Shown here are the first 12 modes. The vertical axis (magnitude) is in a logarithmic
scale. It is evident that the decay of {En, Cn, D n } is faster than exponential, implying
that very few terms are actually contributing to the summation, the higher order terms
are negligible. The coefficient {An} fluctuates at the beginning and decays later as the
order increases. Recall that {An} is always associated with In(z), while the others are
associated with H~l)(z), where z = 0(0.1). Taking this into account, we have found
that only the first 7 modes are significant.
Figure 5 shows the pressure at the center of the fluid as a function of angle of
incidence. The incident wave is a plane compressional wave. The pressure is scaled by
Po = - paw 2 • Three calculations are shown in this figure. The solid line is the result for
an equivalent circular borehole, the long dashed line shows the case where the P-wave
incidents along the minor axis, and the short dashed line shows the case where the
P-wave travels along the major axis. The pressure in the fluid is greater than that in
an equivalent circular borehole when the incident P-wave is parallel to the minor axis;
it is smaller when the incident P-wave is parallel to the major axis. The difference is
about 10% at normal incidence, which is of the same order as the borehole ellipticity.
At grazing incidence, these three curves converge, showing that a P-wave propagating
in the vertical direction cannot 'see' the geometry of the fluid-filied borehole at all.
165
Acoustic wave propagation
Figure 6 shows the pressure at the center of the fluid for a plane SV-wave incidence.
The notations are the same as those in Figure 5. Again, the pressure in the fluid is
larger when the incident SV-wave is parallel to the minor axis, and smaller when the
incident SV-wave is parallel to the major axis. The difference is on the same order as
the borehole ellipticity.
An explanation for these observations is that the elliptical borehole is stiffer against
deformation along the minor axis than along the major axis. This has been shown in
Randall's analysis (Randall, 1991b). He found that the phase velocity of the odd dipole
(polarized in the direction of the minor axis) is larger than that of the even dipole
(polarized in the direction of the major axis).
The solid motion in the formation, however, is affected less by the irregularity of
the fluid-solid boundary. Figure 7 shows both the horizontal and vertical components
of the solid displacement on the borehole wall at an azimuthal angle = 0° (measured
from the major axis). The displacements are scaled by the total displacement of the
incident P-wave. Obviously, the dependence of the solid motion on the direction of the
incident wave is insignificant compared to the effect on the pressure in the fluid. This
observation is also true for a plane SV-wave incidence.
e
Other Borehole Cross-sections
Figure 8 shows a borehole cross-section without a symmetry axis. The dashed line is its
equivalent circular borehole (in the sense of equal surface area). We compute only the
pressure at the center of the fluid (as marked by the solid dot in this figure) for waves
coming from various directions, since the effect of irregularities on the solid displacement
is insignificant in the frequency range we are interested in.
Figure 9 shows the pressure at the center of the fluid as a function of angle of
incidence. The incident wave is a plane P-wave and the formation is Berea sandstone.
A total of 8 curves are shown for the azimuth of the incident wave ranging from 0° to
360° at 45° intervals. The pressure is scaled by Po. As expected, the pressure in the
fluid increases with the increase of the angle of incidence. However, the curves split
into two branches depending on the azimuthal angle of the incident wave. A closer
analysis shows that the curves labeled as 45°, 90° and 225° belong to a cluster with
smaller amplitudes, while the rest are in another cluster with larger amplitudes. The
explanation is as follows. Independent of the irregularity of the borehole, the dipole
motion tends to split into two and only two branches with the faster one polarized
along the effective minor axis and the slower one along the effective major axis. The
predominant contribution to the pressure in the fluid comes from the monopole and
dipole components of the incident wave, and the reminder are negligible.
Figure 10 shows the same calculation as Figure 9 but for a plane SV-wave incidence.
The pressure shows a lobe around 45° incidence. Again, the pressure in the fluid splits
into two branches depending on the azimuthal angle of the incident wave. The one closer
to the effective minor axis of the irregular borehole tends to generate larger pressure in
the fluid; the one closer to the effective major axis tends to induce smaller pressure.
166
Peng and Cheng
APPLICATIONS TO ACOUSTICAL WELL LOGGING
The mode-matching method developed in this paper can also be applied to sonic logging
problems where one is interested in the effects of borehole irregularity on multipole
(especially dipole) logging seismograms. Among the possible effects is the mode splitting
(or cross-mode coupling) in an irregular borehole, a phenomenon that could also be
attributed to formation anisotropy (Hatchell and Cowles, 1992).
A Circular Borehole
Sonic logging in a circular borehole is a subject of intensive studies (among others, Cheng
and Toksiiz, 1981; Toksiiz et al., 1984). We compute synthetic dipole seismograms for
a dipole source in a circular borehole using both the mode-matching and the discrete
wavenumber methods (see Schmitt et a!., 1988 for the latter method). Our objective
is to compare the results of the two methods. This serves as an accuracy test for
the application of the mode-matching method to sonic well logging problems. We use
a model where the borehole is circular in cross-section, the radius is 0.1016 m, and
the well is filled with fresh water. The formation is homogeneous with compressional
velocity 3464 mis, shear velocity 2000 mis, and density 2.0 g/cm 3 • A dipole source is
at the borehole center. The dipole receivers are evenly distributed at offsets between
1.0 m and 4.3 m from the source. Both the source and the receivers are polarized in the
same direction. The source time history is a Ricker wavelet with the central frequency
of 2500 Hz. Figure 11 shows the resulting synthetic seismograms. The solid lines in this
figure are the dipole seismograms computed by the mode-matching method; the dashed
lines are those by the discrete wavenumber method. The agreement between these two
methods is remarkable, although the amplitudes at far offsets differ slightly.
An Elliptical Borehole
In the following examples, we use the elliptical borehole shown in Figure 2. The array
of receivers is located at offsets between 1.0 m and 3.2 m from the source. The source
time function is a Ricker wavelet. The central frequency is 4000 Hz. The formation is
Berea sandstone.
In the first two examples, we use a monopole source and an array of dipole receivers
whose orientation is in the direction of either the major axis (Figure 12) or the minor
axis (Figure 13). The seismograms are evaluated slightly off-axis since the displacement
at the absolute center of the borehole is zero because of symmetry considerations. In
both figures, a low frequency event of large amplitude is clearly visible in the dipole
waveform. Surprisingly, it travels with the phase velocity of the tube wave, suggesting
that it is generated by the cross-mode coupling between the propagating tube wave and
the dipole motion in the presence of borehole ellipticity. The dispersive wave trains
surrounding this tube-wave-like event have the characteristics of the dipole waveform,
and are generated by the tube wave continuously along its path of propagation. At
low frequencies, the dipole motion has the phase speed of the formation shear velocity.
Acoustic wave propagation
167
At high frequencies, it travels much slower than the formation shear speed. Close
comparisons also reveal that the phase velocity ofthe dipole wave in Figure 13 (polarized
along the minor axis) is larger than that in Figure 12 (polarized along the major axis).
This result is in agreement with the theoretical prediction by Randall (1991a). The
amplitude of the seismogram in Figure 13 is about 18% larger than that in Figure 12.
Recall that the borehole ellipticity in these examples is about 10%. This observation
seems to suggest that, for an elliptical borehole, differences in the excitation of the two
perpendicularly polarized dipole modes by a monopole source are of the same order as
the borehole ellipticity.
In the next two examples we use a dipole source and an array of monopole receivers.
The source polarization is in the direction of either the major axis (Figure 14) or the
minor axis (Figure 15). Instead of observing tube waves in the monopole seismograms,
one sees a dispersive wave train with characteristics of a dipole motion. It is generated
by a cross-mode coupling between the traveling dipole energy and the monopole motion
at the receiver location. As expected, the phase velocity at high frequencies is larger in
Figure 15 (polarized along the minor axis) than in Figure 14 (polarized along the major
axis).
In summary, the synthetic examples in this section show that in an elliptical borehole
the cross-mode coupling phenomena are evident in sonic logging seismograms using both
monopole and multipole tools. Depending on the orientation of the dipole source or
dipole receivers, the difference in the phase velocities of the dipole flexural waves at
high frequencies and the excitations of the cross-modes are of the same order as the
borehole ellipticity.
DISCUSSIONS AND CONCLUSIONS
In this paper, we have presented a mode-matching method to compute the pressure in
the fluid and the solid displacement in the formation for an elastic wave impinging upon
an irregular borehole. In this method, the wave fields in both the fluid and the formation
are expressed as a finite summation of mode solutions. The unknown coefficients are
determined by satisfying the boundary conditions at the fluid-solid interface in a least
square sense. The Reichel et al. (1991) algorithm is incorporated into the formulation,
achieving both accuracy and computational speed. This technique requires arithmetic
operations on the order of 0 (J) as compared to the boundary integral equation method
which needs O( J2) arithmetic operations (J is the number of discrete samples on the
boundary). For a circular borehole, this method reduces exactly to the one given in
Chapter 2. For a borehole of any other cross-section, the relative errors in satisfying the
boundary conditions are found to be negligible (of course, it is dependent on the order
of truncation).
The method has also been applied to compute the synthetic sonic logs in a borehole of
elliptical cross-section. Several examples have been given to demonstrate the cross-mode
coupling phenomena. A monopole source excites dipole motions whose propagationai
properties are similar to tube waves, and a dipole source excites monopole motions
Peng and Cheng
168
whose propagational properties are similar to flexural waves.
In summary, we draw the following conclusions:
o The pressure in the fluid is sensitive to the irregularity of the borehole cross-section,
while the solid displacement in the formation is less affected.
• For both P-wave and S-wave incidences, the pressure in an elliptical borehole is
larger than the pressure in an equivalent circular borehole if the incident wave is
along the minor axis. It is smaller than that in an equivalent circular borehole if
the incident wave is along the major axis. The relative difference in the amplitudes
of the pressures is on the same order as the borehole ellipticity.
• In a borehole of arbitrary cross-section, the pressure in the fluid splits into two
distinct branches depending on the azimuthal angles of the incident P- and Swaves. The larger branch is associated with incident waves close to the minor
axis, while the smaller branch is associated with those near the major axis.
• In an elliptical borehole, a centered monopole source excites dipole motions in
the borehole fluid whose characteristics are similar to tube waves. A centered
dipole source excites monopole motions whose characteristics are similar to flexural
waves.
ACKNOWLEDGEMENTS
This work is supported by the Borehole Acoustics and Logging Consortium at M.LT.
REFERENCES
Cheng, C.H., and M.N. Toks6z, 1981, Elastic wave propagation in a fluid-filled borehole
and synthetic acoustic logs, Geophysics, 46, 1042-1053.
Dart, R.L., and M.L. Zoback, 1989, Wellbore breakout stress analysis within the central
and eastern continental United States, The Log Analyst, 30, 12-25.
Du Toit, G.F., 1990, The numerical computation of Bessel functions of the first and
second kind for integer orders and complex arguments, IEEE Trans. Antenna and
Propagation, 38, 1341-1349.
Ellefsen, K.J., 1990, Elastic wave propagation along a borehole in an anisotropic medium,
PhD thesis, M.LT., Cambridge, Massachusetts.
Hatchell, P.J., and C.S. Cowles, 1992, Flexural borehole modes and measurement of
shear-wave azimuthal anisotropy, in 62nd Ann. Intemat. Mtg., Soc. Expl. Geophys.,
Expanded abstmcts, 201-204.
Hilchie, D.W., 1968, Caliper logging-theory and practice, The Log Analyst, 9,3-12.
Acoustic wave propagation
169
Liu, H., and C.J. Randall, 1991, Synthetic waveforms in noncircular boreholes using a
boundary integral equation method, in 61sth Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 843-845.
Norris, A., 1990, The speed of a tube wave, J. Acoust. Soc. Am., 87, 414-417.
Okuno, Y., 1990, The mode-matching method, in Analysis Methods fOT Electromagnetic Wave Propagation, edited by Yamashita, E., Artech House Inc., Norwood,
Massachusetts.
Randall, C.J., 1991a, Modes of noncircular fluid-filled boreholes in elastic formations,
J. Acoust. Soc. Am., 89, 1002-1016.
Randall, C.J., 1991b, Multiple acoustic waveforms in nonaxisymmetric boreholes and
formations, J. Acoust. Soc. Am., 90, 1620-1631.
Reichel, L., G.S. Ammar, and W.B. Gragg, 1991, Discrete least squares approximation
by trigonometric polynomials, Math. Comp., 57, 273-289.
Schoenberg, M., 1986, Fluid and solid motion in the neighborhood of a fluid-filled borehole due to the passage of a low frequency elastic plane wave, Geophysics, 51, 11911205.
ToksOz, M.N., C.H. Cheng, and M.E. Willis, 1984, Seismic waves in a borehole-a review,
in Vertical Seismic Profiling: 14B, M.N. Toksoz and R.R. Stewart (eds.), 256-275,
Geophysical Press.
Zimmerman, R.W., 1986, Compressibility of two-dimensional cavities of various shapes,
J. Appl. Mech., 108, 500-504.
Peng and Cheng
170
APPENDIX A: LISTS OF COEFFICIENTS
We list the coefficients of the stresses and displacements in the harmonic expansions as
follows:
ulf)(r) = afkfJ~(kfr),
vlf\r) = afqnJn(kfr),
wlf)(r) = afikzJn(kfr)
U~4»(r) = akpH~l)' (kpr),
U~{)(r) = _f3k~;'H~l)' (ksr),
U~'lj;) (r) = f3.!£.-nH~l)
(ksr)
.,r
V~4»(r)
=
V~{\r) = -f3.,1£.-nH~l)(ksr), V~'lj;)(r)
a~nH~l)(kpr),
p
W~4»(r) = aikzH~l)(kpr),
"'f3r
W~{)(r) = f3i~H~l)(ksr),
=
f3ksH~l)' (ksr)
W~'lj;)(r) = 0
m(l(r) = -Pfafw 2Jn (kfr)
mhr)
= _pa[(w 2 - 2f32k;)H~1)(kpr) + 2~::~(H~l)' (kpr) - ';;':H~l)(kpr))l
mp(r) = 2Pf33kk;:[H~1)(ksr) + k;rH~l)' (ksr) - k¥:2H~1)(ksr)]
m·:I!)(r) = -2Pf33k;n[klr2H~1)(ksr) - k;rH~l)' (ksr)]
T~4»(r)
=
-2paf32kp2n[klr2H~1)(kpr)
- k\H~l)'
(kpr)]
p
p
T~{)(r) = 2Pf33kk~2 n[k~~2H~1) (ksr) - k;rH~l)' (ksr)]
T~'lj;)(r) = -pf33k;[H~1)(ksr) + k~rH~l)' (ksr) - ;fr~H~l)(ksr)]
Z~4»(r) = 2ipaf32kzkpH~1)' (kpr)
Z~{)(r) = -iPf3*~Jt;)k, H~l)' (ksr)
Z~'lj;\r)
=
13
ipf33kztrnH~1\ksr)
.c~¢)(r) = -pa[w 2 - 2f32(ka 2 - k;)]H~l)(kpr)
.c~{)(r) = -2Pf33i;J(ki - k;)H~l)(ksr)
.c~'lj;)(r)=O
M~¢) (r)
= -pa[(a 2 - 2f32)ka 2H~l)(kpr) -
2f32?rH~l)'
(kpr) + 2f32 k~;2n2 H~l) (kpr)]
p
p
M~'\r) = -2Pf33./i--[?rH~l)' (ksr) - k~;2n2 H~l) (ksr)]
Fef3
If
If
M~1jJ)(r) = 2Pf33k;n[k~~2H~l)(ksr) - k;rH~l)' (ksr)]
N~4»(r)
=
2ipaf32kkk:nH~1)(kpr)
p
(1)
ke2 -2k:
()
N'
./i--nH
(k s r)
n (r)' -- ipf33 , kilT
Fef3
n
N~'lj;)(r)
=
ipf33kzksH~l)' (ksr)
171
Acoustic wave propagation
where, as a reminder,
-
ka
= ~,
kf3
= 71'
kp
=
Vk a
2
-
k~,
ks = Vk/- k~ and
= J~ k~. The vertical wavenumber k z = ka cosO for plane P wave incidence and
f
.
k z = k (3 cos 0 for plane shear wave incidence. 0 is the angle of incidence.
kf
APPENDIX B: DISCRETE LEAST SQUARES APPROXIMATION
BY TRIGONOMETRIC POLYNOMIALS
This appendix summarizes an algorithm developed by Reichel et al. (1991) for the
discrete least squares approximation of a real-valued function by trigonometric polynomials. The function values are given at arbitrary distinct nodes in the interval [0, 27f).
This algorithm has been employed to minimize the errors of satisfying the fluid-solid
interface boundary conditions in a borehole of irregular shape.
This algorithm solves the following minimization problem
Ilf -
til ==
ti
m
(
If(lh) - t(Ok)12w~
) 1/2
= minimum,
(B - 1)
given a set of samples {f( Ok) H'=l of f(O) at distinct nodes {Odr=l in the interval [0, 27f).
{Wnr=l is a set of positive weights and
I
t(O) = ao + 2)aj cosjO + bj sinjO),
j=l
aj, bj E R,
(B - 2)
is a trigonometric polynomial of order I. The problem is to find the coefficients {aj, bj g=o
such that the error in equation (B-1) is as small as possible5 .
Making the substitution z
(Grenander and Szeg6, 1984)
= e iO and noticing that equation (B-2) can be written as
t(O) = z-l p(z),
we can recast the minimization in equation (B-1) as
ti
m
(
(g(Zk) - p(Zk)) (g(Zk) - p(Zk)) w~
) 1/2
minimum,
(B - 3)
where g(z) = zlf(O) is the known data, and p(z) = L:j.:J CjZ j is a polynomial of degree
n with n = 21 + 1. The coefficients {Cj }j.:J are related to the coefficients {aj, bj g=o by
ao = Ci,
aj = Ci+j + Ci-j,
{ bj = i(ci+j - Ci-j).
Sif Wk == 1 and {l'h}k=l are equally sampled between
FFT (the fast Fourier transform).
[0 21t"), the Reichel et al. algorithm reduces to
1
Peng and Cheng
172
Let R = [rjk] be an upper triangular matrix whose nontrivial elements are determined by
k
zk-l
= L rjk 'ljJj_l(Z), 1::; k::; n
(B - 4)
j=1
and let Q = [qkj] be a mxn matrix defined by
qkj = 'ljJj(Zk) Wk,
j = 0, "', n - 1; k = 1, "., m
(B - 5)
where {'ljJj}j,.r/ is the family of orthonormal Szeg6 polynomials6 with respect to the
inner product defined by
The usefulness of the matrices R and Q becomes obvious if we write down the result
(B - 6)
QRc=Dg
obtained by Reichel et al. (1991). Where c = [CO, Cl, .•. , Cn_d T E en is the coefficient
vector solving equation (B-3), and g = [g(ZI), g(Z2), ,." g(zm)]T E em is the data vector.
D = diag[wl, W2, .." wm] is a diagonal matrix whose elements are the positive weights
{wd. That matrix serves as a weighting factor or as the variance ofthe data,
This technique relies upon the speed and accuracy for computing Q * and R -1. The
matrix Q, by definition, has orthonormal columns. That is, Q * Q = I, where the superscript star denotes the Hermitian transpose. Q is obtained by simply evaluating the
Szeg6 polynomials at the set of samples {Zk};;'=I; R -1 is composed of the coefficients of
the Szeg6 polynomials in the power basis {1, z, z2, ..., zn}, and is obtained directly from
the coefficients of the Szeg6 polynomials. The Szeg6 polynomials and its expansions on
the power basis can be efficiently computed by making use of the Szeg6 recurrence relations. The scheme requires only O(mn) arithmetic operations as compared to O(mn 2 )
operations needed for other algorithms, where m is number of nodes (samples), n is the
6The Szego polynomial1/J/z) and its complementary polynomial1/Jj(z) satisfy the following Szego
recurrence relations (Reichel et al., 1991):
1Po(z) = 1/J o(z) = 1/"0,
"H1 1P;+l(z)
O"j+1 ~j+l(Z)
= z1/J;(z) +'YH1 1/J;(z) ,
= Z7j+l 'l/Jj(Z) + Wj(Z)'
where the recurrence coefficients 1'j+1 E C and
0"0
=
O'j+1
=
00
=
j = O,l, ... ,n - 2,
> 0 are determined by
( LW~
) 1/2
k=l
'Y;+l = -
< 1, Z1P;(z) > la;,
2
"H' = (1- b;+,1 )'/2,
Acoustic wave propagation
173
number of unknown coefficients. A detailed account of this technique can be found in
algorithm 3.1 and algorithm 4.1 of Reichel et al. (1991).
The Szego polynomials are dependent on the set of weights {Wk} and the set of
samples {Zk}Z'=l on the unit circle. The merit of the Reichel et al. algorithm is that
the basis functions {'l/1j(Zk)}j=l are adaptive to the shape of the noncircular borehole.
Figure 16 shows a representative Szego polynomial 'l/1s(z) on the unit circle, using the
elliptical borehole in Figure 2 and the discretization as marked in the figure. Recall
that 180 elements are used in discretizing the borehole and the order of truncation is L
=10. The Szego polynomial is similar to a sinusoidal function. However, its amplitude
is modulated by the shape of the borehole: it is larger near the minor axis and smaller
near the major axis of the borehole. Although it is periodic with a 21f periodicity, the
real part is not at a maximum and the imaginary part is not zero at () = 0, 1f, and 21f.
174
Peng and Cheng
z
formation
a
~ p
Figure 1: An irregular and fluid-filled borehole in a homogeneous and elastic formation.
The fluid has a compressional velocity at and a density Pt. The formation has a
compressional velocity a, a shear velocity {3, and a density p. In this figure, (r,li,z)
is a cylindrical coordinate system; (n, b,z) is a local orthogonal coordinate frame in
which the boundary conditions between the fluid-solid interface are satisfied.
Acoustic wave propagation
175
12.0
9.60
7.20
4.80
s
u
2.40
"--'
.~
0.0
~
:>-<-2.40
-4.80
-7.20
-9.60
-12.0
-12.0 -9.60 -7.20 -4.80 -2.40 0.0
2.40
4.80
7.20
9.60
12.0
X axis (em)
Figure 2: An elliptical borehole whose cross-section was determined with a downhole
televiewer (adapted from Ellefsen, 1991). The major axis is 19.10 cm; the minor
axis is 16.9 em. The circumference of the borehole is divided into 180 elements.
176
Peng and Cheng
Radins(em) VP(kmIs) VS(kmIs) RHO(g1em3)
9.55/8.45
formation
1.50
4.206
0.0
2.664
1.00
2.14
Error in Satisfying the Boundary Coudition for Normal Stress
P-Wave Incidence;
Elliptical Borehole;
Frequency = 1000 Hz
6.0E-09
5.4E-09
4. BE-09
4.2E-09
I-<
0
!::
3.6E-09
u:l
0
3.GE-09
......ro
2.4E-09
>
.....
......
0
0:::
1.BE-09
1. 2E-09
6.0E-1O
1. OE-12
0.0
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
90.0
Incidence angle (degree)
Figure 3: The relative error in satisfying the boundary condition as a function of angle
of incidence. Shown is an example for the normal stress in the local coordinates
(n,b,z) at the boundary. The borehole is shown in Figure 2. The horizontal axis is
the angle of incidence. The vertical axis is the relative error in a logarithmic scale.
177
Acoustic wave propagation
U)
c
Q)
'13
:;::
5 -
I
a
1- 1AJ1JJ
Q)
0
()
~
"E
Q)
0
Frequency = 1000 Hz
P wave incidence at 45 degree
Long axis = 2 X 9.55 em; short axis = 2 X 8.45 em
Azimuth of incident wave II long axis
~
- 5 -
•
An
III Bn
•
~ ~
,
,
o
en
0
Dn
- 1a -
c..
0
-15
-
Q)
""0
::J
:!:
c..
-20
E
<tl
-25 -
,
CJ)
.2
-30
a
,
,
1
2
3
4
5
6
7
Harmonic Order
8
,
9
10
11
12
n
Figure 4: Magnitude of the coefficients [An, En, en, Dn]~~ti in equations (2a)-(2d) at
different harmonic orders n. The vertical axis is in a logarithmic scale. The calculation is done at a frequency of 1000 Hz and for a plane P-wave incidence at 6 = 45°
with respect to the borehole axis. The borehole is shown in Figure 2.
178
Peng and Cheng
Radius(cm) VP(kmIs) VS(kmIs) RHO(g/cm3)
9.55/8.45
formation
1.50
0.0
4.206
2.664
1.00
2.14
Elastic Wave Coupling into an Irregular Borehole
P-Wave Incidence;
Elliptical Borehole;
Frequency
=1000 Hz
0.15
Parallel to Minor Axis
Equivalent Circular Borehole
Parallel to Major Axis
0.14
0-----
p..
I
.. , .....
0.11
.'
-
0.09
~
0.06
'"'"
~
0.05
"0
--
0.12
:>.
,.0
----------.
<I)
c<i
U
'"
.////
0.08
'-'
..'
...
/,:~
::l
p..
...."
.'
..
'
,
'." .
;
0.03
0.02
0.0
0.0
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
90.0
Incidence angle (degree)
Figure 5: Pressure at the center of the fluid as a function of angle of incidence. The
incident wave is a plane P-wave whose frequency is 1000 Hz. The formation is Berea
sandstone. The pressure is scaled by the pressure of the incident P-wave. The solid
line in this figure is the pressure in the fluid computed for an equivalent circular
borehole. The line in long dashes is the pressure in the fluid for an incident Pwave propagating parallel to the minor axis. The line in short dashes is that for an
incident P-wave propagating parallel to the major axis.
179
Acoustic wave propagation
Radius(cm) VP(kmls) VS(kmIs) RHO(g1cm3)
9.55/8.45
1.50
0.0
1.00
fonnation
4.206
2.664 2.14
Elastic Wave Coupling into an Irregular Borehole
SV-Wave Incidence;
Elliptical Borehole;
Freqnency; 1000 Hz
0.15
Parallel to Minor Axis
Equivalent Circular Borehole
Parallel to Major Axis
0.14
,--.
0
>l<
0.12
I
0.11
;>,
,D
"0
......
0.09
<l.l
C<l
0.08
'"<l.l
....
;:l
0.06
'"'"
0.05
U
,"
"
,,
>l<
--~--
,"....
..-...'.
.... . ..
.........
..........
,
,,
,
":
, .:"
/
,
' ..,
,
.....
/
'-"'
~
........
;;,;
..
'.
,"
,,
,,
,
/
0.03
I
"
•
...
'\.
"'.
;,"
/:
/:
l'
\,
. \,.
•
/:
I
'
...
"
'
•
•
..
','\.
".'\.
"'.,
.
0.02
.. \
....
..'
0.0
0.0
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
Sl.0
90.0
Incidence angle (degree)
Figure 6: Pressure at the center of the fluid as a function of angle of incidence. The
incident wave is a plane SV-wave whose frequency is 1000 Hz. The formation is
Berea sandstone. The solid line in this figure is the pressure in the fluid computed
for an equivalent circular borehole. The line in long dashes is the pressure in the
fluid for an incident SV-wave propagating parallel to the minor axis. The line in
short dashes is that for an incident SV-wave propagating parallel to the major axis.
Peng and Cheng
180
Radlus(cm) VP(km/s) VS(kmls) RHO(!1cm3)
9.55/8.45
l.SO
0,0
l.Oll
rormation
4.206
2.664 2.14
Elastic Wave Coupling into an Irregular Borehole
P-Wave Incidence;
1.20
S
"
0;
U
1. 06
0.96
~
"""
E
0.84
Elliptical BUJ;ehole;
Frequency = 1000 Hz
. . _"....0'://_-
.••• Parallel to Minor Axis
Equivalent Circular Borehole
0.72
U
"
C.
a
£j
"0
'C
N
0.60
0.48
0.36
0.24
0
:r:
0.12
0.0
0.0
9.0
111.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
90.0
Incidence angle (degree)
1.20
---- Parallel to Minor Axis
Equivalent Circular Borehole
.... Parallel to Major Axis
1.08
S
"
0.96
~
0.84
""
0.72
0;
u
E
"g
0.60
C.
~
Ci
0;
0.48
0.36
U
'f!
>"
0.24
0.12
0.0
0.0
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
90.0
Incidence angle (degree)
Figure 7: Horizontal (a) and vertical (b) components.of the solid displacement at the
borehole wall as a function of angle of incidence. The incident wave is a plane Pwave whose frequency is 1000 Hz. The formation is Berea sandstone. The borehole
is elliptical. The displacements are scaled by the total displacement of the incident
wave. The solid lines in these figures are the displacements in the formation computed for an equivalent circular borehole. The lines in long dashes are those for an
incident P-wave propagating parallel to the minor axis. The lines in short dashes
are those for an incident P-wave propagating parallel to the major axis.
Acoustic wave propagation
181
12.0
9.60
.
~,
7.20
.
4.80
S
2.40
U
'-'
.~ 0.0
~
~
-2.40
-4.80
,.~/
._-_ . . . . _-/ ..
-
.... '.,--
.................
..~
-
......
......... ~
- - - _ .... - . _
..
,,
,,,
.. ,
,,
,,,
/I'
225
- - - -
........ ....
~
~
-
-
,.
-- -\,
-,.,
. ,- .
,,
,,
,
,,
,
--.\-
•~ .'
..-+o;-L.
o :•,
.
.....
,
,,
,,
., .. t._
....
,,
-
,. .
'\
~
". ,
-7.20
-9.60
,
-
" ..
---_
.
"
'-............. ~ ..
135
·/45·········
-12.0
-12.0 -9.60 -7.20 -4.80 -2.40 0.0
2.40
4.80
7.20
9.60
12.0
X axis (em)
Figure 8: A borehole whose cross-section lacks any symmetry axis. The dashed circle
in this figure is the equivalent circular borehole. The arrows are the azimuthal
directions of the incident elastic wave in the formation. The dot at the center of the
figure corresponds to the center of the borehole where the pressure in the fluid is
computed. Also shown are 180 elements discretizing the circumference.
182
Peng and Cheng
Radius(em) VP(kmIs) VS(kmIs) RHO(g/ern3)
1.50
0.0
LOO
formation
4.206
2.664 2.14
*******
Elastic Wave Coupling into an Irregular Borehole
P-Wave Incidence; Very Irregular Borehole;
0.16
odegree
45 degree
90 degree
135 degree
180 degree
225 degree
270 degree
315 degree
0.14
~
0.13
0
0...
1
0.11
:>...
..0
'"d
<l.l
.ro
U
'"
Frequency;;;;: 1000 Hz
0.10
--
.
':-' ;:;
0.08
'-'
~
0.06
'"'"<l.l
....
0.05
;:l
0...
0.03
0.02
0.0
0.0
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
90.0
Incidence angle (degree)
Figure 9: Pressure at the center of the fluid as a function of angle of incidence. The
incident wave is a plane P-wave whose frequency is 1000 Hz. The formation is Berea
sandstone. The borehole is shown in Figure 9. The pressure is scaled by the pressure
of the incident P-wave. Shown in this figure are 8 curves with the azimuthal angle
of the incident wave ranging from 00 to 360 0 in intervals of 45 0 • The pressure in
the irregular borehole splits into two distinct branches, depending on the azimuthal
direction of the incident P-wave with respect to the effective major or minor axis of
the borehole.
183
Acoustic wave propagation
Radius(cm) VP(kmIs) VS(kmls) RHO(g1cm3)
1.50
0.0
1.00
formation
4.206
2.664 2.14
*******
Elastic Wave Coupling into an Irregular Borehole
SV·Wave Incidence; Very Irregular Borehole;
0.16
o degree
45
90
135
180
0.14
~
0.13
0
225
270
1
0..
;;..,
..0
'"ro
Frequency = 1000 Hz
0.11
315
0.10
Q)
..-<
U
0.08
C/1
"-...'
~
0.06
C/1
C/1
0.05
;:l
~
0..
0.03
0.02
0.0
0.0
9.0
18.0
27.0
36.0
45.0
54.0
63.0
72.0
81.0
90.0
Incidence angle (degree)
Figure 10: Pressure at the center of the fluid as a function of angle of incidence. The
incident wave is a plane SV-wave whose frequency is 1000 Hz. The formation is
Berea sandstone. The borehole is shown in Figure 9. A total of 8 curves are shown
in this figure with the azimuthal angle of the incident wave ranging from 0° to 360°
in intervals of 45°. The pressure in the irregular borehole splits into two distinct
branches, depending on the azimuthal direction of the incident SV-wave with respect
to the effective major or minor axis of the borehole.
184
Peng and Cheng
Radius(cm) VP(kmIs) VS(kmIs) RHO(glcm3)
10.16
1.50
0.0
1.00
fonnalion
3.464
2.000 2.00
Sonic Logging in a Circular Borehole
Dipole Source II Dipole Receiver; Central Frequency = 2500 Hz
solid line: Mode-Matching Method; dash line: &:hmitt's Method
SEQ:
Xdisl:
11
4.0
6
2.50
1
1.0
0.0
512.0
1024.0 1536.0 2048.0 2560.0 3072.0 3584.0 4096.0 4608.0 5120.0
Time in Microseconds
Figure 11: Comparison between the synthetic seismograms computed by the modematching method and by the classical discrete wavenumber method. The borehole
is circular with a radius of 0.1016 m. The elastic parameters of the formation
are: a = 3464 mis, fJ = 2000 mis, and p = 2.0 glcma A dipole source and
an array of dipole receivers are used. Twelve (12) receivers are evenly distributed
at offsets between 1.0 m and 4.3 m from the source. The source central frequency
is 2500 Hz. The solid lines in this figure show the seismograms computed by the
mode-matching method. The dashed lines are the ones computed by the discrete
wavenumber method. The program for the discrete wavenumber method was written
by D. P. Schmitt (1988).
Acoustic wave propagation
185
Radius(em) VP(kmIs) VS(kmIs) RHO(g1cm3)
9.55/8.45
1.50
0.0
1.00
formation
4.206
2.664
2.14
Sonic Logging in an Elliptical Borehole
Monopole Source; Dipole Receiver II Long Axis; Central Frequency = 4000 Hz
Offset (m),
SEQ:
10
2.80
5
1. 80
1
1. 00
0.0
409.6
819.2
1228.8 1638.4 2048.0 2457.6 2867.2 3276.8 3686.4 4096.0
Time in Microseconds
Figure 12: Synthetic seismograms in an elliptical borehole for a monopole source and
an array of dipole receivers. The dipole receivers are polarized along the major axis
of the borehole shown in Figure 2. The formation is Berea sandstone. The central
frequency of the source is 4000 Hz. Twelve (12) receivers are evenly distributed
at source-receiver offsets be];ween 1.0 m and 3.2 m. The maximum amplitude is
0.833E+02.
Peng and Cheng
186
Radius(cm) VP(kmIs) VS(kmls) RHO(glcm3)
9.55/8.45
1.50
0.0
1.00
formation
4.206
2.664 2.14
Sonic Logging in an Elliptical Borehole
Monopole Source; Dipole Rec:e:iver II Short Axis; Central Frequency = 4000 Hz
SEQ:
Offset (m):
10
2.80
5
1. 80
~rvY'-~------1
1. 00
1
0.0
409.6
819.2
1228.8 1638.4 2048.0 2457.6 2867.2 3276.8 3686.4 4096.0
Time in Microseconds
Figure 13: Synthetic seismograms in an elliptical borehole for a dipole source and an
array of monopole receivers. The dipole source is polarized along the major axis
of the borehole shown in Figure 2. The formation is Berea sandstone. The central
frequency of the source is 4000 Hz. Twelve (12) receivers are evenly distributed at
offsets between 1.0 m and 3.2 m from the source. The maximum amplitude is 0.136.
187
Acoustic wave propagation
Radius(em) VP(kmIs) VS(kmIs) RHO(g1cm3)
9.55/8.45
fonnation
1.50
4.206
0.0
2.664
1.00
2.14
Sonic Logging in an Elliptical Borehole
Source in Long axis; Dipole Source •• Monop<lle Receiver; Central Frequency = 4000 Hz
SEQ:
1---'-----1---'-----'---.1.-----'-----'--,---1--1-----1
Offset (m):
10
2.80
5
1. 80
1
1. 00
0.0
409.6
819.2
1228.8 1638.4 2048.0 2457.6 2867.2 3276.8 368M 4096.0
Time in Microseconds
Figure 14: Synthetic seismograms in an elliptical borehole for a dipole source and an
array of monopole receivers. The dipole source is polarized along the minor axis
of the borehole shown in Figure 2. The formation is Berea sandstone. The central
frequency of the source is 4000 Hz. Twelve (12) receivers are evenly distributed at
offsets between 1.0 m and 3.2 m from the source. The maximum amplitude is 0.150.
188
Peng and Cheng
Radius(cm) VP(km/s) VS(km/s) RHO(glcm3)
9.55/8.45
1.50
0.0
1.00
4.206
2.664 2.14
fonnation
Sonic Logging in an Elliptical Borehole
Source in short axis; Dipole Source -- Monopole Receiver; Central Frequency = 4000 Hz
Offset (m):
SEQ:
10
2.80
5
1. 80
1
1. 00
0.0
409.6
819.2
1228.8 1638.4 2048.0 2457.6 2867.2 3276.8 3686.4 4096.0
Time in Microseconds
Figure 15: The Szegii polynomial >Ps(z) on the unit circle: (top) real part and (bottom)
imaginary part. It is computed for the elliptical borehole shown in Figure 2.